A $p$-adic Eisenstein measure for vector-weight automorphic forms

Abstract

We construct a $p$-adic Eisenstein measure with values in the space of vector-weight $p$-adic automorphic forms on certain unitary groups. This measure allows us to $p$-adically interpolate special values of certain vector-weight $C^{\infty }$ automorphic forms, including Eisenstein series, as their weights vary. This completes a key step toward the construction of certain $p$-adic $L$-functions.

We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz’s $p$-adic families of Eisenstein series for Hilbert modular forms.